Limits of integration

In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral $\int_{a}^{b} f (x) d x$

of a Riemann integrable function $f$ defined on a closed and bounded interval are the real numbers $a$ and $b$ , in which $a$ is called the lower limit and $b$ the upper limit. The region that is bounded can be seen as the area inside $a$ and $b$ .

For example, the function $f (x) = x^{3}$ is defined on the interval $[2, 4]$ $\int_{2}^{4} x^{3} d x$ with the limits of integration being $2$ and $4$ .

Integration by Substitution (U-Substitution)

In Integration by substitution, the limits of integration will change due to the new function being integrated. With the function that is being derived, $a$ and $b$ are solved for $f (u)$ . In general, $\int_{a}^{b} f (g (x)) g^{'} (x) d x = \int_{g (a)}^{g (b)} f (u) d u$ where $u = g (x)$ and $d u = g^{'} (x) d x$ . Thus, $a$ and $b$ will be solved in terms of $u$ ; the lower bound is $g (a)$ and the upper bound is $g (b)$ .

For example, $\int_{0}^{2} 2 x \cos (x^{2}) d x = \int_{0}^{4} \cos (u) d u$

where $u = x^{2}$ and $d u = 2 x d x$ . Thus, $f (0) = 0^{2} = 0$ and $f (2) = 2^{2} = 4$ . Hence, the new limits of integration are $0$ and $4$ .

The same applies for other substitutions.

Improper integrals

Limits of integration can also be defined for improper integrals, with the limits of integration of both $lim_{z \to a^{+}} \int_{z}^{b} f (x) d x$ and $lim_{z \to b^{-}} \int_{a}^{z} f (x) d x$ again being a and b. For an improper integral $\int_{a}^{\infty} f (x) d x$ or $\int_{- \infty}^{b} f (x) d x$ the limits of integration are a and ∞, or −∞ and b, respectively.

Definite Integrals

If $c \in (a, b)$ , then $\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x .$

References

↑ "31.5 Setting up Correct Limits of Integration". math.mit.edu. Retrieved 2019-12-02.
↑ "𝘶-substitution". Khan Academy. Retrieved 2019-12-02.
↑ "Calculus II - Improper Integrals". tutorial.math.lamar.edu. Retrieved 2019-12-02.
↑ Weisstein, Eric W. "Definite Integral". mathworld.wolfram.com. Retrieved 2019-12-02.

[1] "31.5 Setting up Correct Limits of Integration". math.mit.edu. Retrieved 2019-12-02.

[2] "𝘶-substitution". Khan Academy. Retrieved 2019-12-02.

[3] "Calculus II - Improper Integrals". tutorial.math.lamar.edu. Retrieved 2019-12-02.

[4] Weisstein, Eric W. "Definite Integral". mathworld.wolfram.com. Retrieved 2019-12-02.

Limits of integration

Integration by Substitution (U-Substitution)

Improper integrals

Definite Integrals

See also

References